A family of 'etale coverings of the affine line, by Kirti Joshi

This paper has appeared in J. Number Theory 59 (1996), no. 2, 414-418, and so the dvi version has been removed. In this note we show that one can use Drinfel'd modular curves to construct a family of etale coverings of the affine line. This leads to construction of a profinite quotient of the algebraic fundamental group of the affine line. Our result is proved by using the moduli of Drinfel'd $A$-modules of rank two over $C$ with $I$-level structure (where $C$ is a suitably large field ); these Drinfel'd modular curves give rise to a tower of galois coverings of the affine line ramified only at one point (on the linei ), and the ramification is tame and independent of $I$. The tame ramification can be removed in the entire tower by invoking a suitable variant of Abhyankar's lemma; this variant of Abhyankar's lemma also calculates the Galois group in our context. A similar construction leads to a tower of coverings of the affine space of any dimension. This note will appear in Journal of Number Theory.

Kirti Joshi <kirti@motive.math.tifr.res.in>